Let $R$ and $S$ be rings and $M$ be an $S$-$R$- bimodule. Then consider the triangular matrix ring constructed from these data (as, e.g., in Auslander-Reiten-Smalo). what one can say about the projective dimension of a (right) module $(X, Y)_f$ over this triangular matrix ring? Maybe I should explain that, in this situation, $X$ is a right $S$-module, $Y$ is a right $R$-module and $f: X\otimes_S M\longrightarrow Y$ is $R$-linear.
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